**Classification of Triangles**

Triangles the meaning of it is in its own name. Triangle= tri + angle, means the geometrical figure having three angles in it with closed shape and it is possible only when that geometrical shape has three sides which encloses the three angle. Thus, triangle is the geometrical shape having three angle with three sides.

The above figure shows the triangle ABC with three sides as AB, BC, CD and three angles as <A, <B and <C.

We have seen from 1^{st} std. the shapes of geometrical figure like circle, triangle, square and many. These shapes are having different properties with physical meaning also, and they are classified in different ways.

The triangle is the basic shape which is classified on the basis of sides of triangles and also on the basis of angles of triangles. The detailed classification of triangles with their properties is explained as follows. We not have seen the geometrical shapes in geometry but in our nature also there different geometrical shaped living organisms are also present only we have to see them through the vision of curiosity of learning.

**Properties of triangles:**

- Triangle has three sides and three angles.
- The sum of all the three internal angles in a triangle must be 180
^{0} - The sum of any two sides of triangle is always greater than the third side.

**Classification of triangles on the basis of their sides:**

On the basis of sides of triangles they are classified as follows:

- Equilateral triangle
- Isosceles triangle
- Scalene triangle

**1.) Equilateral triangle:**

The equilateral triangle is the triangle which is having all the three sides in equal length.

If the one side measure the length as “a” then other two sides of equilateral triangle measure the same side length as “a”.

The following figure shows the equilateral triangle with three sides AB=BC=CD= a.

**Properties of equilateral triangle:**

- The equilateral triangle has three sides having equal length.
- It has the three internal angles which are congruent to each other.
- And hence, in equilateral triangle each internal angle measures as 60
^{0}. - As it measure all sides equal in length and all angles are congruent, it is also called as regular polygon.
- The perpendiculars which are drawn from each of the vertex of the equilateral triangle bisect the side in front of it.

**2) Isosceles triangle:**

Isosceles triangle is the triangle having two sides equal in length and the third side measures different length than these two sides.

The following figure shows the isosceles triangle whose two sides AB and AC are equal in length let us suppose “a”, and third side BC measures different length as “b”.

**Properties of Isosceles triangle**:

- It has two sides equal in length and third is different in length than these two equal sides.
- The internal angles opposite to the sides which are equal in length are congruent.
- Here, sides AB and AC are equal in length and hence the angles opposite to them i.e. <ABC and <ACB are congruent.
- If the isosceles triangle is right angled triangle then one of the internal angle is 90
^{0}, and remaining each angle measures 45^{0}, so that the sum of internal angles must be 180^{0}.

**3) Scalene triangle**:

Scalene triangle is the triangle which is having all the sides different in length i.e. unequal. And hence all the internal angles are also different.

The following figure shows the Scalene triangle ABC, in which sides AB, BC and AC all are different in length and hence the angles <BAC, <ABC and <ACB all are different.

**Properties of Scalene triangle**:

- It has all the three sides different in length i.e. unequal and hence all the three internal angles are also different.
- The internal angles may be acute, obtuse or right angled.

**Classification of triangles on the basis of their internal angle:**

On the basis of internal angles the triangles are classified into three types as given below.

- Acute triangle
- Obtuse triangle
- Right triangle

**1) Acute triangle:**

Acute triangle is the triangle which having all three internal angles less than 90^{0}.

The three sides of acute angle may be same or different.

The equilateral triangle is the best example of acute triangle because all its internal angles are measures as 60^{0}, and also three sides are equal in length as shown in fig1.

The isosceles triangle is also the acute triangle. The figure 2 shows the acute isosceles triangle in which the internal angles opposite to the equal side measures as 70^{0} and the remaining third angle measures as 40^{0}.

The scalene triangle also may be an acute triangle. The figure 3 shows scalene acute triangle in which the three internal angle measures as 40^{0}, 70^{0} and 80^{0}.

**Properties of Acute triangle:**

- It has all three internal angles less than 90
^{0}. - Equilateral triangle, isosceles triangle and scalene triangle are also the acute triangles which are having all three internal angles less than 90
^{0}.

**2) Obtuse Triangle:**

The obtuse triangle is the triangle which is having only one obtuse anglei.e. greater than 90^{0} and the remaining two internal angles are acute.

The following figure shows the obtuse angled triangle ABC, in which angle <ABC is obtuse and the other two angles <CAB and <BCA are acute.

**Properties of obtuse triangle:**

- It has only one internal angle as obtuse angle i.e. greater than 90
^{0}and other two are acute angles i.e. less than 90^{0}. - The side opposite to the obtuse angle is the longest side of the obtuse triangle.
- As one of the internal angle is obtuse, then the sum of other two internal angles must be less than 90
^{0}.

**3) Right triangle:**

Right triangle is the triangle which has one of the internal angle as 90^{0}i.e. right angled and hence the sum of remaining two internal angles must be 90^{0}.

The following figure 1 shows the right angled triangle in which <ACB is the right angle.

The isosceles triangle having one right angle is also the right triangle as shown in figure 2.

**Properties of right triangle:**

- It has one right angle i.e.90
^{0}in measure. - It has the sum of other two internal angles as 90
^{0}. - The side opposite to right angle is called as hypotenuse and it is the longest side.
- Every right angles triangle obeys Pythagoras Theorem